They would. But it would be very gradual. The spikes would grow a microscopic amount each day as the cells replicate, slowly forcing the sharp tip against the eyeball until the pressure is finally enough to puncture it. By that point, though, the doctors would realize what was happening and probably surgically remove the eye so the patient wouldn't die from complications.

probably not, at least not for very long. now i don't know about the rest of the body, but the brain actually keeps functioning for about 7 minutes after the heart stops beating, so with that in mind, i assume the bone spike things would keep growing for that same amount of time, until the brain finally shut off and with it all other functions. after that i think it's safe to assume that the bone spikes will stop growing. now i'm not claiming to be an expert on this subject, i'm just giving my personal opinion on the current topic

Well cancer cells grow autonomously from the brain. Some could even say they have a "mind of their own," so to speak. The heart is what pumps blood, and therefore vital nutrients, to the cancer cells, so it really depends on whether or not the heart is still functioning, not the brain.

wait, so if i were to have taken this skull and placed it into a vat of liquid which was able to mimic blood, in the respect of providing nutrients and oxygen, then theoretically the bone spikes would continue to grow as if nothing happened?

If you were able to instantly transfer it as soon as the blood flow stops and make sure that there's still an uninterrupted flow to the tissues, then possibly. But I don't see how that would be possible. Once blood flow stops, the cells die. Once the cells die, they can't start growing again.

Not for more than a few minutes. All cells, normal or cancerous, need fresh oxygen and nutrients through a blood supply. If the person dies, they're no longer eating or breathing and the blood is no longer circulating, so the cancer cells would die with the rest of the body.